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I see four quadratic functions listed in the image. Let's break each one down and analyze or solve them as needed:

### 15. \( f(x) = x^2 + 8x + 12 \)
This is a standard quadratic equation in the form \( ax^2 + bx + c \). To find the roots (if needed), you can factor the quadratic, use the quadratic formula, or complete the square.

### 16. \( f(x) = x^2 + 5x + 6 \)
Another quadratic equation in the form \( ax^2 + bx + c \). You can similarly factor or apply the quadratic formula to solve for \(x\).

### 17. \( f(x) = -x^2 + 8x - 19 \)
This is a quadratic equation with a negative leading coefficient, indicating the parabola opens downwards. Solving this equation can be done through factoring (if possible) or the quadratic formula.

### 18. \( f(x) = -3x^2 + x - 1 \)
This quadratic equation has a larger negative leading coefficient. Again, you can solve this using the quadratic formula, as factoring may not work easily here.

Would you like me to solve any of these equations, find their roots, or provide another type of analysis?

Here are five follow-up questions for practice:
1. How would you factor \( f(x) = x^2 + 8x + 12 \)?
2. What is the vertex of the parabola given by \( f(x) = -x^2 + 8x - 19 \)?
3. How does the negative leading coefficient affect the shape of the parabola for \( f(x) = -3x^2 + x - 1 \)?
4. Can you use the quadratic formula to solve \( f(x) = x^2 + 5x + 6 \)?
5. What are the x-intercepts of \( f(x) = -x^2 + 8x - 19 \)?

**Tip:** When factoring quadratics, always check if the discriminant (from the quadratic formula) is a perfect square. This can help determine if the equation is easily factorable.

Solve or analyze the following quadratic equations: 1) f(x) = x^2 + 8x + 12, 2) f(x) = x^2 + 5x + 6, 3) f(x) = -x^2 + 8x - 19, 4) f(x) = -3x^2 + x - 1

Explore step-by-step solutions for various quadratic equations, including f(x) = x^2 + 8x + 12 and f(x) = -x^2 + 8x - 19. Learn key algebraic concepts and how to apply the quadratic formula to find roots. Suitable for students in Grades 9-12.

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